| Type | Proposer | Responder | Payouts |
|---|---|---|---|
| A | L | D | 3, 0 |
| A | L | U | 2, 1 |
| A | R | D | 4, 1 |
| A | R | U | 1, 1 |
| B | L | D | 3, 0 |
| B | L | U | 3, 1 |
| B | R | D | 3, 2 |
| B | R | U | 0, 4 |
444 Lecture 23
2024-04-09
This week’s assignment is about a signaling game.
Figure 1: Tree for Weekly 5 practice version
| Type | Proposer | Responder | Payouts |
|---|---|---|---|
| A | L | D | 3, 0 |
| A | L | U | 2, 1 |
| A | R | D | 4, 1 |
| A | R | U | 1, 1 |
| B | L | D | 3, 0 |
| B | L | U | 3, 1 |
| B | R | D | 3, 2 |
| B | R | U | 0, 4 |
In this tree, Proposer has four possible strategies:
And Responder has four possible strategies
So the strategy table has four rows for Proposer, and four columns for Responder, and that means 16 cells.
If there was no randomness, then for any pair, you could tell what each player would get.
But there is randomness. So what do we do?
I’ll start with the answers, then work back to how to figure them out.
| P1 | DD | DU | UD | UU |
|---|---|---|---|---|
| LL | 3, 0 | 3, 0 | 2.4, 1 | 2.4, 1 |
| LR | 3, 0.8 | 1.8, 1.6 | 2.4, 1.4 | 1.2, 2.2 |
| RL | 3.6, 0.6 | 1.8, 0.6 | 3.6, 1 | 1.8, 1 |
| RR | 3.6, 1.4 | 0.6, 2.2 | 3.6, 1.4 | 0.6, 2.2 |
Let’s start in the bottom right, the 0.6, 2.2.
The formula for P1’s expected return is:
Let’s start in the bottom right, the 0.6, 2.2.
The formula for P2’s expected return is:
If A happens, and P1 plays RR, that means P1 chooses R (the first letter in P1’s strategy).
If P1 chooses R, and P2 plays UU, that means P2 chooses U (the second letter in P1’s strategy).
So the row of the table we’re on is where A, then R, then U happen.
| Type | Proposer | Responder | Payouts |
|---|---|---|---|
| A | L | D | 3, 0 |
| A | L | U | 2, 1 |
| A | R | D | 4, 1 |
| A | R | U | 1, 1 |
| B | L | D | 3, 0 |
| B | L | U | 3, 1 |
| B | R | D | 3, 2 |
| B | R | U | 0, 4 |
And at that row P1 gets 1, and P2 gets 1.
If B happens, and P1 plays RR, that means P1 chooses R (the first letter in P1’s strategy).
If P1 chooses R, and P2 plays UU, that means P2 chooses U (the second letter in P1’s strategy).
So the row of the table we’re on is where B, then R, then U happen.
| Type | Proposer | Responder | Payouts |
|---|---|---|---|
| A | L | D | 3, 0 |
| A | L | U | 2, 1 |
| A | R | D | 4, 1 |
| A | R | U | 1, 1 |
| B | L | D | 3, 0 |
| B | L | U | 3, 1 |
| B | R | D | 3, 2 |
| B | R | U | 0, 4 |
And at that row P1 gets 0, and P2 gets 4.
So P1’s expected return is:
And P2’s expected return is:
Let’s do one more of these: the one where Player 1 plays RL and Player 2 plays UD.
The formulae are still the same, but we have to be a bit more careful about the applications.
As the table says, the results are 3.6 and 1.0. Let’s work to get there.
The formula for P1’s expected return is:
If A happens, and P1 plays RL, that means P1 chooses R (the first letter in P1’s strategy).
If P1 chooses R, and P2 plays UD, that means P2 chooses D (the second letter in P1’s strategy).
So the row of the table we’re on is where A, then R, then D happen.
| Type | Proposer | Responder | Payouts |
|---|---|---|---|
| A | L | D | 3, 0 |
| A | L | U | 2, 1 |
| A | R | D | 4, 1 |
| A | R | U | 1, 1 |
| B | L | D | 3, 0 |
| B | L | U | 3, 1 |
| B | R | D | 3, 2 |
| B | R | U | 0, 4 |
And at that row P1 gets 4, and P2 gets 1.
If B happens, and P1 plays RL, that means P1 chooses L (the second letter in P1’s strategy).
If P1 chooses L, and P2 plays UD, that means P2 chooses U (the first letter in P1’s strategy).
So the row of the table we’re on is where B, then L, then U happen.
| Type | Proposer | Responder | Payouts |
|---|---|---|---|
| A | L | D | 3, 0 |
| A | L | U | 2, 1 |
| A | R | D | 4, 1 |
| A | R | U | 1, 1 |
| B | L | D | 3, 0 |
| B | L | U | 3, 1 |
| B | R | D | 3, 2 |
| B | R | U | 0, 4 |
And at that row P1 gets 3, and P2 gets 1.
So P1’s expected return is:
And P2’s expected return is
Now we have to find the equilibria for the game.
An equilibria is a pair of choices where neither party can do better by changing their view.
So the top right corner: LL, UU is an equilibrium.
If P1 plays LL, then P2 gets (in expectation)
They can’t do better than playing UU. (Ties aren’t a problem.)
If P2 plays UU, then P1 gets (in expectation)
They can’t do better than playing LL.
We don’t want to have to check all 16 cells this way. Fortunately, we can speed things up.
There are a few approaches - here’s one.
Go through each column, and find the highest score for P1.
That will be P1’s best response to that strategy for P2.
Note that there may be more than one of these, if there are ties.
Then for each of the things you find, check whether P2 can do better by switching.
If so, it’s not an equilibria.
If not, it is one.
And you’ll find all the equilibria this way.
| P1 | DD | DU | UD | UU |
|---|---|---|---|---|
| LL | 3, 0 | 3, 0 | 2.4, 1 | 2.4, 1 |
| LR | 3, 0.8 | 1.8, 1.6 | 2.4, 1.4 | 1.2, 2.2 |
| RL | 3.6, 0.6 | 1.8, 0.6 | 3.6, 1 | 1.8, 1 |
| RR | 3.6, 1.4 | 0.6, 2.2 | 3.6, 1.4 | 0.6, 2.2 |
Applying this technique shows that LL,UU and RL,UD are the only equilibria.
An equilibrium is pooling if P1 does the same thing in each state. So P1 plays LL or RR.
An equilibrium is separating if P1 does different things when the states are different. So P2 plays LR or RL.
If we were doing more game theory, there are more questions we could ask at this point.
Some equilibria that you can find this way are not intuitively sensible.
Sometimes that’s because they imply P2 will do things that no longer make sense once they see P1’s move. The notion of sequential equilibrium was developed to rule out these.
For real life cases, there are times when it is sensible to bluff, but not sensible to stick with the bluff once it has been exposed. But the procedure we’ve outlined here might include equilibria where committing to the bluff even when the cards are shown might be an equilibrium.
And into the really not on the exam stuff, there is the question of what to do with the equilibrium in the US-UK game where P1 believes that P2 believes that P1 will buy tea if and only if American.
It is somewhat less clear that this creats real life problems.
Lots of data: https://www.oecd-ilibrary.org/education/education-at-a-glance-2023_e13bef63-en
Relative wage by educational attainment
For most of these facts we can get very resilient data cross-nationally.
But there’s one group, which will be important in what follows, for which we really have to focus on US data.
That’s the ‘some college’ group.
Note the gap between ‘some college’ and ‘associate degree’ (which means at least associate).
To the best of my knowledge, this gap is many times higher in the US than in any other rich country.
I’m not going to go into the gender splits in these numbers, but I think the rough picture is this.
The college premium is particularly high for men.
Not breaking things down by gender tends to under-state the premium because more women go to college and women earn less.
We should include as contrasts things that are really far from reality - part of the job is to explain why those things are far from reality.
Ideally, we’d like explanations where people are basically rational.
Maybe we’ll have to give up on the latter, but let’s try.
People go to college because it means they get more money than not going, and people like money.
Two problems:
What do you think?
Why do people go to college, and why do employers pay so much for college grads?
All of these models explain the existence of a college wage premium.
They aren’t completely incompatible - the truth may contain some measure of each.
We’ll go over next time much more detail about which of them explain the various facts.
Just getting into college isn’t valuable:
Doing 2-3 years of college isn’t valuable:
The college wage premium rises over time:
Much more to follow!